# Volume 65, № 5, 2013

### Fibonacci lengths of all finite $p$-groups of exponent $p^2$

Ukr. Mat. Zh. - 2013. - 65, № 5. - pp. 603–610

The Fibonacci lengths of finite p-groups were studied by Dikici and coauthors since 1992. All considered groups are of exponent $p$ and the lengths depend on the Wall number $k(p)$. The p-groups of nilpotency class 3 and exponent $p$ were studied in 2004 also by Dikici. In the paper, we study all $p$-groups of nilpotency class 3 and exponent $p^2$. Thus, we complete the study of Fibonacci lengths of all $p$-groups of order $p^4$ by proving that the Fibonacci length is $k(p^2)$.

### Analog of the John theorem for weighted spherical means on a sphere

Savost’yanova I. M., Volchkov V. V.

Ukr. Mat. Zh. - 2013. - 65, № 5. - pp. 611–619

We study generalizations of the class of functions with zero integrals over the balls of fixed radius. An analog of the John uniqueness theorem is obtained for weighted spherical means on a sphere.

### On 3-dimensional *f*-Kenmotsu manifolds and Ricci solitons

Ukr. Mat. Zh. - 2013. - 65, № 5. - pp. 620–628

The aim of the present paper is to study 3-dimensional *f*-Kenmotsu manifolds and Ricci solitons. First, we give an example of a 3-dimensional *f*-Kenmotsu manifold. Then we consider a Riccisemisymmetric 3-dimensional *f*-Kenmotsu manifold and prove that a 3-dimensional *f*-Kenmotsu manifold is Ricci semisymmetric if and only if it is an Einstein manifold. Moreover, we investigate an η-parallel Ricci tensor in a 3-dimensional *f*-Kenmotsu manifold. Finally, we study Ricci solitons in a 3-dimensional *f*-Kenmotsu manifold.

### Asymptotic estimates for the solutions of boundary-value problems with initial jump for linear differential equations with small parameter in the coefficients of derivatives

Kasymov K. A., Nurgabyl D. N., Uaissov A. B.

Ukr. Mat. Zh. - 2013. - 65, № 5. - pp. 629–641

We establish asymptotic estimates for the solutions of singularly perturbed boundary-value problems with initial jumps.

### Lebesgue-type inequalities for the de la Vallée-poussin sums on sets of entire functions

Ukr. Mat. Zh. - 2013. - 65, № 5. - pp. 642–653

For functions from the sets *C* ^{ψ} _{β} *L* _{ s }, 1 ≤ *s* ≤ ∞,
where ψ(*k*) > 0 and \( {\lim_{{k\to \infty }}}\frac{{\psi \left( {k+1} \right)}}{{\psi (k)}} \) , we obtain asymptotically sharp estimates for the norms of deviations of the de la Vallée-Poussin sums in the uniform metric represented in terms of the best approximations of the (ψ, β) -derivatives of functions of this kind by trigonometric polynomials in the metrics of the spaces *L* _{ s }. It is shown that the obtained estimates are sharp on some important functional subsets.

### Global nonexistence of solutions for a system of nonlinear viscoelastic wave equations with degenerate damping and source terms

Bayoud M., Ouchenane D., Zennir Kh.

Ukr. Mat. Zh. - 2013. - 65, № 5. - pp. 654–669

The global existence and nonexistence of solutions for a system of nonlinear wave equations with degenerate damping and source terms supplemented with initial and Dirichlet boundary conditions was shown by Rammaha and Sakuntasathien in a bounded domain Ω ⊂ \( {{\mathbb{R}}^n} \) , *n* = 1, 2, 3, in the case where the initial energy is negative. A global nonexistence result on the solution with positive initial energy for a system of viscoelastic wave equations with nonlinear damping and source terms was obtained by Messaoudi and Said-Houari. Our result extends these previous results. We prove that the solutions of a system of wave equations with viscoelastic term, degenerate damping, and strong nonlinear sources acting in both equations at the same time are globally nonexisting provided that the initial data are sufficiently large in a bounded domain Ω of \( {{\mathbb{R}}^n} \) , *n* ≥ 1, the initial energy is positive, and the strongly nonlinear functions f_{1} and f_{2} satisfy the appropriate conditions. The main tool of the proof is based on the methods used by Vitillaro and developed by Said-Houari.

### Constructive description of monogenic functions in a three-dimensional harmonic algebra with one-dimensional radical

Plaksa S. A., Pukhtaevich R. P.

Ukr. Mat. Zh. - 2013. - 65, № 5. - pp. 670–680

We present a constructive description of monogenic functions that take values in a three-dimensional commutative harmonic algebra with one-dimensional radical by using analytic functions of complex variable. It is shown that monogenic functions have the Gâteaux derivatives of all orders.

### Li–Yorke sensitivity for semigroup actions

Ukr. Mat. Zh. - 2013. - 65, № 5. - pp. 681–688

We introduce and study the concept of Li–Yorke sensitivity for semigroup actions (dynamical systems of the form (*X*, *G*), where *X* is a metric space and *G* is a semigroup of continuous mappings of this space onto itself). A system (*X*, *G*) is called *Li–Yorke sensitive* if there exists positive ε such that, for any point *x* ∈ *X* and any open neighborhood *U* of this point, one can find a point *y* ∈ *U* for which the following conditions are satisfied:

(i) *d*(*g*(*x*), *g*(*y*)) > ε for infinitely many *g* ∈ *G*,

(ii) for any δ > 0; there exists *h* ∈ *G* satisfying the condition *d*(*h*(*x*), *h*(*y*)) < δ.

In particular, it is shown that a nontrivial topologically weakly mixing system (*X*, *G*) with a compact set *X* and an Abelian semigroup G is Li–Yorke sensitive.

### On polymer expansions for generalized Gibbs lattice systems of oscillators with ternary interaction

Ukr. Mat. Zh. - 2013. - 65, № 5. - pp. 689–697

We propose a new short proof of the convergence of high-temperature polymer expansions in the thermodynamic limit of canonical correlation functions for classical and quantum Gibbs lattice systems of oscillators interacting via pair and ternary potentials and nonequilibrium stochastic systems of oscillators interacting via a pair potential with Gibbsian initial correlation functions.

### Linear Combinations of the Volterra Dissipative Operator and Its Adjoint Operator

Gubreev G. M., Olefir E. I., Tarasenko A. A.

Ukr. Mat. Zh. - 2013. - 65, № 5. - pp. 706–711

We study the spectral properties of linear combinations of the Volterra dissipative operator and its adjoint operator in a separable Hilbert space.

### Main Inverse Problem for Differential Systems With Degenerate Diffusion

Ibraeva G. T., Tleubergenov M. I.

Ukr. Mat. Zh. - 2013. - 65, № 5. - pp. 712–716

The separation method is used obtain sufficient conditions for the solvability of the main (according to Galiullin’s classification) inverse problem in the class of first-order Itô stochastic differential systems with random perturbations from the class of Wiener processes and diffusion degenerate with respect to a part of variables.

### Probability Measures on the Group of Walsh Functions With Trivial Equivalence Class

Il’inskaya I. P., Neguritsa D. S.

Ukr. Mat. Zh. - 2013. - 65, № 5. - pp. 717–721

We establish necessary and sufficient conditions for the retrieval, to within a shift, of a composition of three Poisson distributions and a uniform distribution on five or six elements of the group of Walsh functions according to the absolute values of their characteristic functions.

### Cross Topology and Lebesgue Triples

Karlova O. O., Mykhailyuk V. V.

Ukr. Mat. Zh. - 2013. - 65, № 5. - pp. 722–727

The cross topology γ on the product of topological spaces *X* and *Y* is the collection of all sets *G* ⊆ *X* × *Y* such that the intersections of *G* with every vertical line and every horizontal line are open subsets of the vertical and horizontal lines, respectively. For the spaces *X* and *Y* from a class of spaces containing all spaces \( {{\mathbb{R}}^n} \) , it is shown that there exists a separately continuous function *f* : *X* × *Y* → (*X* × *Y*, γ) which is not a pointwise limit of a sequence of continuous functions. We also prove that each separately continuous function is a pointwise limit of a sequence of continuous functions if it is defined on the product of a strongly zero-dimensional metrizable space and a topological space and takes values in an arbitrary topological space.

### One Property of Ring *Q*-Homeomorphisms With Respect to a *p*-Module

Ukr. Mat. Zh. - 2013. - 65, № 5. - pp. 728–733

We establish sufficient conditions for a ring *Q*-homeomorphisms in \( {{\mathbb{R}}^n} \) , *n* ≥ 2, with respect to a *p*-module with *n* − 1 < *p* < *n* to have the finite Lipschitz property. We also construct an example of the ring *Q*-homeomorphism with respect to a *p*-module at a fixed point which does not have the finite Lipschitz property.

### Fixed-Point Theorems and Common Fixed-Point Theorems on Spaces Equipped With Vector-Valued Metrics

Hosseinzadeh H., Jabbari A., Razani A.

Ukr. Mat. Zh. - 2013. - 65, № 5. - pp. 734–740

We show the existence of fixed points and common fixed points for single-valued generalized contractions on the spaces equipped with vector-valued metrics.